The equations of motion for the extended theory can be derived from the action [1]}
\(S_E[q,p,\gamma ,\sigma ] = \int _{0}^{T} dt \Bigl \lbrace p_i \dot{q}_i- H_E - \sigma _1 \, {\cal C}_1^{(sc)} - \sigma _2\, {\cal C}_2^{(sc)} \Bigr \rbrace  \ ,\) 
